Formal Proof Systems Reveal Overlooked Ambiguities in Advanced Mathematics

Table Of Links

Abstract

1. Acknowledgements & Introduction

2. Universal properties

3. Products in practice

4. Universal properties in algebraic geometry

5. The problem with Grothendieck’s use of equality.

6. More on “canonical” maps

7. Canonical isomorphisms in more advanced mathematics

8. Summary And References

Summary And References

Whilst I am not making any claims about errors in the literature or holes in arguments which cannot be filled in after some work, I am arguing that these holes do exist, and that new mathematics might need to be done by formalisers in order to fill in these holes in an efficient way. The holes are of two kinds. Firstly there is the issue of people making constructions or proving theorems which make essential use of a model of a mathematical object which is defined up to unique isomorphism; the hole here is that it needs to be checked that the argument does not depend on the explicit details of the model.

\ Mathematicians are well aware of this when it comes to, say, picking a basis for a vector space and then checking that nothing important depended on the choice, or picking a representative for an equivalence class and then checking that nothing important depends on the representative. However they seem to be less careful when doing more advanced mathematics, confusing “a” localisation with “the” localisation or “a” pullback with “the” pullback, and leaving to the reader the details of checking that many diagrams commute.

\ One useful trick is to abuse the equality symbol, making it mean something which it does not mean; this can trick the reader into thinking that nothing needs to be checked. Sometimes such checks can be surprisingly painful, and it may be easier to restructure a mathematical argument than to actually make these checks. The second kind of hole is the issue of various maps (like boundary maps in exact sequences) being regarded as “canonical” where now they are in fact not unique, and there are implicit choices of sign being made.

\ Unfortunately it is not at all difficult to point to explicit examples in the literature where an author does not state precisely which convention they are using when it comes to things like the theory of Shimura varieties, or homological algebra. This puts an unnecessary burden on the careful mathematician (for example Conrad, or a computer theorem prover) who is attempting to use or verify the work. Both of these issues have shown up in my formalisation work, and I expect them to show up more often as we go deeper into the formalisation of modern mathematics.

References

[AX23] David Kurniadi Angdinata and Junyan Xu, An Elementary Formal Proof of the Group Law on Weierstrass Elliptic Curves in Any Characteristic, 14th International Conference on Interactive Theorem Proving (ITP 2023) (Dagstuhl, Germany) (Adam Naumowicz and Ren´e Thiemann, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 268, Schloss Dagstuhl – Leibniz-Zentrum f¨ur Informatik, 2023, pp. 6:1– 6:19.

[BCM20] Kevin Buzzard, Johan Commelin, and Patrick Massot, Formalising perfectoid spaces, Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, New Orleans, LA, USA, January 20-21, 2020 (Jasmin Blanchette and Catalin Hritcu, eds.), ACM, 2020, pp. 299–312.

[BPL21] Anthony Bordg, Lawrence Paulson, and Wenda Li, Grothendieck’s schemes in algebraic geometry, March 2021, https://isa-afp.org/entries/Grothendieck_Schemes.html, Formal proof development.

[Buz] Kevin M. Buzzard, Grothendieck’s approach to equality, https://www.youtube.com/watch?v=-OjCMsqZ9ww, Accessed: 12-08-2023. [Buz19] Buzzard, Kevin, The inverse of a bijection, 2019, [Online; accessed 12-Aug-2023].

[Con00] Brian Conrad, Grothendieck duality and base change, Lecture Notes in Mathematics, vol. 1750, Springer-Verlag, Berlin, 2000. MR 1804902

[dFF23] Mar´ıa In´es de Frutos-Fern´andez, Formalizing Norm Extensions and Applications to Number Theory, 14th International Conference on Interactive Theorem Proving (ITP 2023) (Dagstuhl, Germany) (Adam Naumowicz and Ren´e Thiemann, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 268, Schloss Dagstuhl – LeibnizZentrum f¨ur Informatik, 2023, pp. 13:1–13:18.

[Gro60] A. Grothendieck, El´ements de g´eom´etrie alg´ebrique. I. Le langage des sch ´ ´emas, Inst. Hautes Etudes Sci. Publ. Math. (1960), no. 4, 228. MR 217083 ´

[Lan97] R. P. Langlands, Representations of abelian algebraic groups, Pacific J. Math. (1997), 231–250, Olga Taussky-Todd: in memoriam. MR 1610871

[Liv23] Amelia Livingston, Group Cohomology in the Lean Community Library, 14th International Conference on Interactive Theorem Proving (ITP 2023) (Dagstuhl, Germany) (Adam Naumowicz and Ren´e Thiemann, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 268, Schloss Dagstuhl – Leibniz-Zentrum f¨ur Informatik, 2023, pp. 22:1–22:17.

[mC20] The mathlib Community, The lean mathematical library, Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, ACM, jan 2020.

[Mil80] James S. Milne, Etale cohomology ´ , Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531

[Sta18] The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu, 2018.

[Wei59] Andr´e Weil, Correspondence [signed “R. Lipschitz”], Ann. of Math. (2) 69 (1959), 247–251, Attributed to A. Weil. MR 100637 [Wik04a] Wikipedia contributors, Monoidal category — Wikipedia, the free encyclopedia, 2004, [Online; accessed 12-Aug-2023].

[Wik04b] , Ordered pair — Wikipedia, the free encyclopedia, 2004, [Online; accessed 20- May-2023].

[Zha23] Jujian Zhang, Formalising the Proj Construction in Lean, 14th International Conference on Interactive Theorem Proving (ITP 2023) (Dagstuhl, Germany) (Adam Naumowicz and Ren´e Thiemann, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 268, Schloss Dagstuhl – Leibniz-Zentrum f¨ur Informatik, 2023, pp. 35:1– 35:17.

[ZM23] Max Zeuner and Anders M¨ortberg, A univalent formalization of constructive affine schemes, 2023. Email address: k.buzzard@imperial.ac.uk Department of Mathematics, Imperial College London

:::info Author: KEVIN BUZZARD

:::

:::info This paper is available on arxiv under CC BY 4.0 DEED license.

:::

\